Neuro -fuzzy-networks-for-identification-of-mathematical-model-parameters-of-geofield

World Academy of Science, Engineering and Technology
International Journal of Mathematical, Computational Science and Engineering Vol:1 No:12, 2007

Neuro – Fuzzy Networks for Identification of
Mathematical Model Parameters of Geofield
A. Pashayev, R. Sadiqov, C. Ardil, F. Ildiz , and H. Karabork
m

identification of parameters for mathematical models of geofields is
proposed and checked. The effectiveness of that soft computing
technology is demonstrated, especially in the early stage of
modeling, when the information is uncertain and limited.

International Science Index 12, 2007 waset.org/publications/8427

Keywords—Identification, interpolation methods, neurofuzzy networks, geofield.

F

I. INTRODUCTION

many problems in sciences on Earth (geodesy,
geology, geophysics, cartography, photogrammetry, etc.)
the problem of modeling the geofields surface (height, depth,
pressure, temperature, pollution factor, etc.), wich is usually
displayed on maps by means of isolines, is urgent. If
representation of geofields surface is possible as function of
two variables h=f (x, y), which has hi values at (xi, yi), (i
OR

= 1, n ) peaks, the digital model of this function is required for
computer processing and storage.
We are going to consider the digital model of geofield
(DMG) as a set of digital values of continuous objects in
cartography (e.g. height of a relief) for which their spatial
coordinates and the mean of structural description are
specified. It will allow calculating the values of geofield in the
given area. The important part of any DMG is the method of
interpolating of its surface. For this, various ways of
interpolation yield various results which can be estimated only
from the point of view of practical applications [1- 6].
Nowadays, more than ten methods of surface interpolation
are known. They are as fellows algebraic and orthogonal
polynoms, rational fractions; in some eases they take functions
satisfying some apriori given conditions (e.g. positivity of f (x,
y)) values; multi squadric function, at which approximation is
reached bu means of square – law functions (squadric),
representing hyperboles; splines; geostatic methods (kriging).
However, none of them is completely universal. We shall
consider widely used procedure of interpolation by algebraic
polynoms

n

h ( x, y) = ∑∑ A ijxiyj

Abstract—The new technology of fuzzy neural networks for

i =0 j=0

where i = 0, m; j = 0, n - exponents; A - factors at
decomposition members received on a method of least squares
(LSM).
Realization of these methods is rather simple; therefore they
have received a wide circulation [1-5]. This is the linear
interpolation modeling of a surface as set of triangles. Thus
the normal to a surface is constant along all surface of a
triangle and sharply varies at transition through the sides
separating triangles. Therefore, LSM constructed with use of
linear interpolation, frequently insufficiently adequately
represent the investigated phenomenon [2].
The much better result (absence of sharp differences of
values of researched parameter, smoothness of isolines), is
given by modeling with the use of polynomial to interpolation
of higher degree. The general (common) expression for
calculation of value, for example, heights h in a point of a
surface with coordinates (x, y) looks like:
m

h(x,y)= ∑

m− j

∑

Cjkxj yk

(1)

j= 0 k =0

We shall consider a special case (1) at m=2, that is the
equation of regress of the second order
H(x,y)=C00+C10x+C01y+C20x2+C11xy+C02y2

(2)

The equation of measurements of target coordinate h for
this case will be written down as:
Zh=C00+C10x+C01y+C20x2+C11xy+C02y2+δh
Then the model of an experimental material can be
presented in the following matrix kind:
Zh=Xθ+δh,
where Zh = || z1h, z2h,…, znh || - a vector of measurements of
target coordinate h; θ= ||C00, C10, C01, C20, C11, C02||T - a vector
of required factors;

Manuscript received June 30, 2005. This work (R. Sadiqov) was supported
by TUBITAK NATO-PC B program.
Authors are with the National Academy of Aviation, AZ1045, Bina, 25th
km, Baku, Azerbaijan (corresponding author to provide phone: 99412- 49728-29, fax: 99412-497-28-29, e-mail : sadixov@mail.ru).

312


1 x1
1 x2
X=
− −

y1
y2

1 xn

yn

2
x1 x1 y1
x 2 x 2 y2
2
− − −

x2
n

x n yn

neurons. When an neural network is used to solve equation
(3), the input signals of the network are the fuzzy values of the
~
~
variable B = (~ , ~ ), and the output is H . The fuzzy values of
x y
the parameters ~
are the network parameters. We present
c

2
y1
y2
2
−

y2
n

jk

A structural matrix; n - quantity(amount) of points of
supervision (measurements).
Usually for identification (estimation) of factors of a
polynom (2) are used LSM of the following kind
θ=(ХТХ)-1(ХТZh),
Dθ=(ХТХ)-1σ2,


where D θ - dispersive matrix of mistakes of estimations.
The use of statistical probability methods, such as the leastsquares method, requires preliminary analysis of the data for
normality of the sample distribution. A normality check
assumes that the following four conditions are satisfied.
1. The intervals x ± σ, x ± 2σ x and x ± 3σ must contain 68,
95, and 100%, respectively, of the sample values x is the
mean and о is the standard deviation).
2. The coefficient of variation V must not exceed 33%.
3. The kurtosis E x and the asymmetry coefficient S k must

be close to zero.
4. x ≈ M . where M is the sample median.
The analysis [6] was used for modeling (2) showed that
distribution contradicted the normality assumption (Table 1).
It must be noted that in the early stage modeling of
geofield, the data are not only limited and uncertain but also
fuzzy (the output and input coordinates of the system are
measured in definite intervals and their values are measured
with errors).

It is therefore necessary to identify the parameters of a
mathematical model of a multivariate fuzzy object described
by the regression equation
m m− j

(3)

j= 0 k = 0

( j = 0, m; k = 0, m, j + k ≤ m)

where ~ jk
c

⎧
⎪1 − ( x − x ) / α, if
x − α < x < x;
⎪
µ( x ) = ⎨
x < x < x + β;
⎪1 − ( x − x ) / β, if
⎪
0,
otherwise
⎩

Neural-network training is the principal task in solving the
c
problem of identification of the parameters ~ jk of equation
(3). An α -section is used to train the parameter values [7].
We assume the presence of experimentally obtained fuzzy
statistical data. From the input and output data we compose
~~
training pairs for the network (B, T) . To construct a model of a
~

process, the input signals B are fed to the neural network
input (Fig.1); the output signals are compared with standard

~

output signals T .
After comparison, the deviation is calculated:
~ 1 l ~ ~
E = ∑ (H i − Ti ) 2
2 i =1

When an α -section is used. the deviations for the left and
right parts are calculated by the formulas
l

E1 =

1
2

∑ [h i1 ( α ) − t i1 ( α ) ]2 ,

E2 =

1
2

∑ [h i 2 ( α ) − t i 2 ( α ) ]2 ,

i =1
l

i =1

E = E1 + E 2 ,

II. PROBLEM FORMULATION AND SOLUTION

~
H m = ∑ ∑ ~ jk ⊗ ~ j ⊗ ~ k
c
x
y

the fuzzy variables in triangular form, the membership
functions of which are calculated by the formula

where

~
H i (α) = [h i1 (α), h i 2 (α)] ;

Training (correction) of the network parameters is
concluded when the deviations E for all training pairs are less
than the specified value (Fig. 2). Otherwise, it is continued
until E is minimized.
The network parameters for the left and right parts are
corrected a-s follows:

are the desired fuzzy parameters.

We shall determine the fuzzy values of the parameters ~ jk
c

cn 1 = co 1 + γ
jk
jk

of equation (3) using. experimental fuzzy statistical data of the
~
process, i.e., the input ~ , ~ and output H coordinates of the
x y
model. Let us consider a solution of this problem using fuzzy
logic and neural networks [7,8].
A neural network consists of interconnected sets of fuzzy

~
Ti (α) = [t i1 (α), t i 2 (α)]

∂E
,
∂c jk

Here co 1 , c n 1 , co 2 and
jk
jk
jk

cn 2 = co 2 + γ
jk
jk

cn 2
jk

∂E
,
∂c jk

(4)

are the old and new values

of the left and right pans of the neural network parameters
~ = [ c , c ] , and γ is the training rate.
cjk
jk1 jk2

313


h 51 = c111x 2 y2 ; h52 = c112x1y1 , and the correction formulas was

III. NUMERICAL EXAMPLE
Large Let us consider the mathematical model is described
the equation of fuzzy a regression (consider a special case (3)
at m=2):
~
H = ~00 + ~10~ + ~01~ + ~20~2 + ~11~ ~ + ~02~2.
c
c x c y c x c xy c y
(5)

We shall construct a neural structure for solution of (5) in
which the network parameters are the coefficients
~ , ~ , ~ , ~ , ~ , ~ . The structure has four inputs and one
c00 c10 c01 c20 c11 c02
output (Fig. 3).
Using a neuro-network structure, we employ (4) to train the
network parameters. For a = 0 , we obtain the following
expressions:

performed.
The network parameters were thus trained using the
described fuzzy-neural network structure and experimental
data. As a result, network-parameter values that satisfied the
experimental statistical data were found (see Table 2):
~ = (1.4124 1.4223 1.4275
c00
;
;
);
~ = (1.98842.11312.2339
c
;
;
);
10

~ = (−2.5353− 2.5349− 2.5326
c01
;
;
);
~ = (−1.1043−1.1042−1.1036
c
;
;
)
20

~ −(−0.8845−0.8741−0.8639
c11
;
;
);
~ = (1.31581.31621.3166
c
;
;
).
02

l
∂E 2
= ∑(h i2 − t i2 );
∂c002 i=1

l
∂E1
= ∑(h i1 − t i1 )x1;
∂c101 i=1


l
∂E1
= ∑(h i1 − t i1 );
∂c001 i=1

l
∂E 2
= ∑(h i2 − t i2 )x 2 ;
∂c102 i=1

l
∂E1
= ∑(h i1 − t i1 )y1;
∂c011 i=1

l
∂E 2
= ∑(h i2 − t i2 )y 2 ;
∂c102 i=1

l
∂E1
2
= ∑(h i1 − t i1 )x1 ;
∂c111 i=1

l
∂E2
= ∑(h i2 − t i2 )x 2 ;
2
∂c112 i=1

l
∂E1
= ∑(h i1 − t i1 )x1y1;
∂c201 i=1

l
∂E2
= ∑(h i2 − t i2 )x 2 y2
∂c202 i=1

These data were obtained as a result of 20-minute training
of the neural network.The coefficients ~00 , ~10 , ~01, ~20 , ~11, ~02
c c c c c c
regression equation (5) were evaluated by a program written
in Turbo Pascal on an IBM PC.

l
∂E1
2
= ∑(h i1 − t i1 )y1 ;
∂c021 i=1

l
∂E1
= ∑(h i1 − t i1 )x 2 y2 ;
∂c111 i=1

For a = 1, we obtain
l
∂E3
= ∑(h i3 − t i3 );
∂c003 i=1

l
∂E3
= ∑(h i3 − t i3 )x 3 ;
∂c103 i=1

∂E3 l
=∑hi3 −ti3)y3;
(
∂c013 i=1

IV.

(6)

l
∂E2
= ∑(h i2 − t i2 )y2
2
∂c202 i=1

The use of fuzzy neural networks (Soft Computing) to solve
problems that involve evaluation parameters of mathematical
models of geofields advantages over traditional statisticalprobability approaches. Primary is the fact that the proposed
procedure can be used regardless of the type of distribution of
the parameters geofield. The more so because, in the early
stage of modeling, it is difficult to establish the type of
parameter distribution, due to insufficient data.
REFERENCES

l
∂E2
= ∑(h i2 − t i2 )x1y1;
∂c112 i=1

[1]
[2]

l
∂E3
= ∑(h i3 − t i3 )x 3 y3 ;
∂c113 i=1

[3]
[4]

l
∂E3
2
= ∑(h i3 − t i3 )x 3
∂c 203 i=1

∂E3 l
2
=∑hi3 −ti3)y3
(
∂c023 i=1

CONCLUSIONS

[5]

(7)

As a result of training (6) and (7), we find network
parameters that satisfy the knowledge base with the required
training quality.
Fuzzy statistical data (see Table 2) were collected from
experiments before the computer simulation It should be noted
that for negative values of the parameter ~jk (~jk < 0) , the
c c

[6]
[7]
[8]

formulas that include the parameter ~jk in (5) and the
c

correction of that parameter in (6) will have changed forms.
For example, if ~jk < 0 , the formula for the fifth expression,
c
which includes ~jk in (5) will have the following form:
c

314

M. Yanalak, Height interpolation in digital terran models. Ankara, Harita
dergisi, Temmuz 2002. Sayi: 128. p. 44 – 58.
A.Berlyant, L. Ushakova, Cartographic animations. Moscow: Scientific
World, 2000.
M. Jukov, S. Serbenyuk and V. Tikunov, Mathematical – cartographig
modeling in geography. Moscow: misl, 1980.
O. Akima, P. Hiroshi, Bivariate interpolation and smooth surface fitting
for irregulary distributed date points. ACM Transactions for
Mathematical Software. June 1978. p. 148 – 159.
J. Delhhome, Kriging in the hudro sciences // Adv. Water Res. 1978.
Vol. 1.№5.
L.Buryakovskii, I. Dzhafarov and Dzhefanshir, Modeling systems.
Moscow: Nedra, 1990.
R. Yager, L. Zadeh. (Eds.) Fuzzy sets, Neural Networks and Soft
Computing. Van Nostrand Reinhold – New York, 1994.
H. Mohamad. Fundamentals of Artificial Neural Networks, MIT Press,
Cambridge, Mass., London, 1995


APPENDIX
TABLE I
NORMALITY ASSUMPTION
68%

0.71≠0.59
non – exe –
cution

95%

100%

V<33%

Ex→0

Sk→0

77.7%
execu tion

x≈M

91.6 %
non – exe –
cution

100%
execu tion

47 %
non – exe –
cution

0.45
non – exe –
cution

1.14
non – exe cution

~
B

~
T

Input-output
relation
(knowledge base)


Scaler

~
H

Нечеткая
Fuzzy
neuralНС
etwork

?+

~
Ε

Scaler

-

Fig. 1 Neural identification system

Correction algorithm

~

B

Input
signals

Neural
network

Parameters

Random-number
generator

Deviations

Training
quality

Fig. 2 System for network-parameter training (with backpropagation)

315

Target
signals

~

H


~ x
~ y ~ x
~ y
c10 ~ + c 01 ~ + c 20 ~ 2 + c 02 ~ 2

~
x

~
c00

~
x

~
x

~
H

~2
x

~
y

~
y
~
y

~ x y
c11~ ~

~2
y

Fig. 3 Structure of neural network for second-order regression equation


TABLE II
THE EXPERIMENTAL STATISTICAL DATA

~
y

3,7,11

17,21,25

31,35,39

45,49,53

59,63,67

73,77,81

0.77,0.81,0.85
0.48,0.52,0.56
0.37,0.41,0.45
0.30,0.34,0.38
0.27,0.31,0.35
0.23,0.27,0.31

1.08,1.13,1.17
0.68,0.72,0.76
0.53,0.57,0.61
0.43,0.47,0.51
0.39,0.43,0.47
0.34,0.38,0.42

1.28,1.33,1.44
0.81,0.85,0.89
0.63,0.67,0.71
0.52,0.58,0.60
0.46,0.50,0.54
0.41,0.45,0.49

1.43,1.47,1.51
0.89,0.93,0.97
0.69,0.73,0.77
0.57,0.61,0.65
0.51,0.55,0.59
0.46,0.50,0.54

1.49,1.53,1.57
0.93,0.97,1.01
0.72,0.76,0.60
0.60,0.64,0.68
0.54,0.58,0.62
0.47,0.51,0.55

1.48,1.50,1.54
0.91,0.95,0.99
0.71,0.75,0.79
0.59,0.63,0.67
0.53,0.57,0.61
0.47,0.51,0.55

~
x
28,31,35
50,54,58
68,72,76
82,86,90
92,96,100
96,100,104

316

Neuro -fuzzy-networks-for-identification-of-mathematical-model-parameters-of-geofield

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